Multi-fuzzy Subgroups - Semantic Scholar

Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 8, 365 - 372

Multi-fuzzy Subgroups Sabu Sebastiana,1 and T. V. Ramakrishnan Department of Mathematical Sciences, Kannur University Mangattuparamba, Kerala-670567, India [email protected] Abstract Multi-fuzzy set theory is an extension of fuzzy set theory, which deals with the multi dimensional fuzziness. In this paper we introduce the concepts of multi-fuzzy subgroups and normal multi-fuzzy subgroups.

Mathematics Subject Classification: MSC: 20N25; 03E72; 08A72 Keywords: Multi-fuzzy set; Multi-fuzzy subgroup; Normal multi-fuzzy subgroup

1

Introduction

We introduced the theory of multi-fuzzy sets [6,7] in terms of multi-dimensional membership functions and investigated some properties of multilevel fuzziness. Theory of multi-fuzzy set is an extension of theories of fuzzy sets [10] and L-fuzzy sets [2]. Complete characterization of many real life problems can be done by multi-fuzzy membership functions of the objects involved in the problem. Rosenfeld [5] started fuzzifiction of various algebraic concepts by his paper Fuzzy groups. Das [1] studied the interrelationship between the fuzzy subgroup and its level subsets. Liu [3], Mukharjee and Bhattacharya [4] proposed the concept of normal fuzzy sets. In this paper we extend the basic concepts of group theory into multi-fuzzy sets.

2

Preliminaries

Throughout this paper, we will use the following notations. X and Y stand for universal sets, the sets I, J and K stand for indexing sets, L and M stand for partially ordered sets, {Lj : j ∈ J} and {Mi : i ∈ I} are families of complete lattices with order reversing involutions, unless it is stated otherwise and LX stands for the set of all functions from X to L. The products 1

Corresponding author

366

Sabu Sebastian and T. V. Ramakrishnan

Mi , k ∈ K.

Lj ,

Nk ,

MiX ,

LYj and

NkZ varies over i ∈ I, j ∈ J and

Definition 2.1. [2] Let X be a nonempty ordinary set and L be a complete lattice. An L-fuzzy set on X is a mapping A : X → L, that is, the family of all the L-fuzzy sets on X is just LX consisting of all the mappings from X to L. Definition 2.2. [8] Let : M → M and : L → L be order reversing involutions. A mapping h : M → L is called an order homomorphism, if it satisfies the conditions h(0) = 0, h(∨ai ) = ∨h(ai ) and h−1 (b ) = (h−1 (b)) . h−1 : L → M is defined by ∀b ∈ L, h−1 (b) = ∨{a ∈ M : h(a) ≤ b}. Wang [8] proved the following properties of order homomorphism. For every a ∈ M and p ∈ L; a ≤ h−1 (h(a)), h(h−1 (p)) ≤ p, h−1 (1L ) = 1M , h−1 (0L) = 0M and a ≤ h−1 (p) if and only if h(a) ≤ p if and only if h−1 (p ) ≤ a . Both h and h−1 are order preserving and arbitrary join preserving maps. More over h−1 (∧ai ) = ∧h−1 (ai ). Proposition 2.3. [9] Let f : L1 → L2 be a union preserving mapping. If f is injective, then f −1 (f (a)) = a, ∀a ∈ L1 and if f is surjective, then f (f −1 (b)) = b, ∀b ∈ L2 .

2.1

Multi-fuzzy sets

Definition 2.4. [6,7] Let X be a nonempty set, J be an indexing set and {Lj : j ∈ J} a family of partially ordered sets. A multi-fuzzy set A in X is a set : A = {x, (μj (x))j∈J : x ∈ X, μj ∈ LX j , j ∈ J}. The function μA = (μj )j∈J is called the multi-membership function of the multi-fuzzy set A. If |J| = n, a natural number, then n is called the dimension of A. Complement of A is A = {x, (μj (x))j∈J : x ∈ X}, where μj is the order reversing involution of μj . Definition 2.5. [6] Let {Lj : j ∈ J} be a family of partially ordered sets, A = {x, (μj (x))j∈J : x ∈ X, μj ∈ LX j , j ∈ J} and B = {x, (νj (x))j∈J : X x ∈ X, νj ∈ Lj , j ∈ J} be multi-fuzzy sets in a nonempty set X with the product order, then A B if and only if μj (x) ≤ νj (x), ∀x ∈ X and ∀j ∈ J The equality, union and intersection of A and B are defined as: (a) A = B if and only if μj (x) = νj (x), ∀x ∈ X and ∀j ∈ J; (b) A B = {x, (μj (x) ∨ νj (x))j∈J : x ∈ X};

367

Multi-fuzzy subgroups

(c) A B = {x, (μj (x) ∧ νj (x))j∈J : x ∈ X}. Proposition 2.6. Let A, B, C ∈ MiX be any multi-fuzzy sets in X then: (a) A A = A, A A = A ; (b) A A B, B A B, A B A and A B B; (c) A B if and only if A B = B if and only if A B = A; X Proposition 2.7. [7] Let A∈ Mi and for any α ∈ Mi , the set A[α] = {x ∈ X : A(x) ≥ α, α ∈ Mi } be the α-level of A. If A, B ∈ MiX , then for every α, β ∈ Mi : (a) α ≤ β implies A[β] ⊆ A[α] ; (b) A B if and only if A[α] ⊆ B[α] ; (c) A = B if and only if A[α] = B[α] .

2.2

Multi-fuzzy extensions of functions

Definition 2.8. [7] Let f : X → Y and h : Mi → Lj be a functions. The multi-fuzzy extension of f and the inverse of the extension are X Y Y X −1 f : Mi → Lj and f : Lj → Mi defined by f (A)(y) =

h(A(x)), A ∈

MiX , y ∈ Y

y=f (x)

and f −1 (B)(x) = h−1 (B(f (x))), B ∈

LYj , x ∈ X;

Mi → Lj is where h−1 is the upper adjoint [8] of h. The function h : called the bridge function of the multi-fuzzy extension of f . Mi → Theorem 2.9. [7] If an order homomorphism h : Lj is the bridge function for the multi-fuzzy extension of a crisp function f : X → Y, X Y then for any k ∈ K, Ak ∈ Mi , Bk ∈ Lj : (a) f (0X ) = 0Y ; (b) A1 A2 implies f (A1 ) f (A2 ); (c) f (Ak ) = f (Ak );

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(d) f ( Ak ) f (Ak ); (e) f (A[α] ) ⊆ f (A)[h(α)]; (f ) f −1 (1Y ) = 1X and f −1 (0Y ) = 0X ; (g) B1 B2 implies f −1 (B1 ) f −1 (B2 ); (h) f −1 (Bk ) = f −1 (Bk ); (i) f −1 ( Bk ) = f −1 (Bk ); (j) (f −1 (B)) = f −1 (B ); (k) A f −1 (f (A)); (l) f (f −1 (B)) B.

are injective Theorem 2.10. [7] (i) If f : X → Y and h : Mi → Lj −1 −1 Mi ), then (that is f (f (x)) = x, ∀x ∈ Xand h (h(m)) = m, ∀m ∈ −1 X f (f (A)) = A, for every A ∈ Mi . Moreover the multi-fuzzy extension f : MiX → LYj is injective. (ii) If f : X →Y and h : Mi → Lj are surjective, then f (f −1 (B)) = B, for every B ∈ LYj .

3

Multi-fuzzy groups

In this section we introduce the concepts of multi-fuzzy subgroup and multifuzzy subgroupoid. Partial order ≥ is the opposite order relation of the partial order ≤ . Definition 3.1. A multi-fuzzy set A of a group G is called a multi-fuzzy subgroup of G if (i) A(x) ∧ A(y) ≤ A(xy), and (ii) A(x−1 ) = A(x), for all x, y ∈ G. Equivalently, a multi-fuzzy set A is called a multi-fuzzy subgroup of G, if A(x) ∧ A(y) ≤ A(xy −1), ∀x, y ∈ G. It follows immediately from this definition that A(x) ≤ A(e), for all x ∈ G, where e is the identity element of G. If A is a multi-fuzzy subgroup of a group G, then for x, y ∈ G, A(xy −1 ) = A(e) implies A(x) = A(y). A multi-fuzzy subset A of a group G is called a multi-fuzzy subgroupoid of G, if A(x) ∧ A(y) ≤ A(xy), for all x, y ∈ G. A multi-fuzzy subgroupoid A of a group G is a multi-fuzzy subgroup of G, if A(x−1 ) = A(x) for all x, y ∈ G.


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Theorem 3.2. If {Ai : i ∈ I} is a family of multi-fuzzy subgroups of a group G, then Ai is a multi-fuzzy subgroup of G. Proof. Let A = Ai . For any x, y ∈ G; −1 −1 A(xy −1 ) = ( A i )(xy ) = Ai (xy ) ≥ (Ai (x) ∧ Ai (y)) = ( Ai (x)) ∧ ( Ai (y)) = ( Ai )(x) ∧ ( Ai )(y) = A(x) ∧ A(y).

Proposition 3.3. Union of two multi-fuzzy subgroups of a group G need not be a multi-fuzzy subgroup of G. Proof. Straight forward. Theorem 3.4. Let G1 and G2 be groups, f be a group homomorphism from G1 into G2 and a finite meet preserving order homomorphism h : Mi → Lj be the bridge function for the multi-fuzzy extension of f . If A be a multifuzzy subgroup of G1 , then f (A) is a multi-fuzzy subgroup of G2 . Proof. Let x, y ∈ G2 . If either f −1 (x) or f −1 (y) is empty, then f (A)(x) ∧ f (A)(y) = 0 and implies f (A)(xy −1) ≥ f (A)(x) ∧ f (A)(y). Assume that neither f −1 (x) nor f −1 (y) is empty. Therefore there exist u ∈ f −1 (x) and v ∈ f −1 (y) such that A(u) = {A(t) : t ∈ G1 , x = f (t)} and A(v) = f (u)(f (v))−1 = f (uv −1 ) and {A(t) : t ∈ G1 , y = f (t)}. We have xy −1 = so uv −1 ∈ f −1 (xy −1 ). Thus f (A)(xy −1) = {h(A(z)) : z ∈ G1 , xy −1 = −1 )) ≥ h(A(u) ∧ A(v)) f (z)} ≥ h(A(uv = h(A(u)) ∧ h(A(v)) =h( {A(t) : x = f (t)}) ∧ h( {A(t) : y = f (t)}) = ( {h(A(t)) : x = f (t)}) ∧ ( {h(A(t)) : y = f (t)}) = f (A)(x) ∧ f (A)(y). Hence f (A) is a subgroup of G2 . Theorem 3.5. Let G1 and G2 be groups, 1 → G2 be a group homo f :G morphism and order homomorphism h : Mi → Lj be the bridge function for the multi-fuzzy extension of f . If B be a multi-fuzzy subgroup of G2 , then f −1 (B) is a multi-fuzzy subgroup of G1 . Proof. Note that h−1 : Lj → Mi is order preserving and meet preserving [8], and f −1 (B)(y) = h−1 ◦ B(f (y)). For any x, y ∈ G; f −1 (B)(xy −1 ) = h−1 ◦ B(f (xy −1)) = h−1 ◦ B(f (x)(f (y))−1) ≥ h−1 [B(f (x)) ∧ B(f (y))] = h−1 [B(f (x))] ∧ h−1 [B(f (y))] = (f −1 (B)(x)) ∧ (f −1 (B)(y)). Theorem 3.6. Let f be an injective group homomorphism from group G1 into group G2 , an injective meet preserving order homomorphism h : Mi → Lj be the bridge function for the multi-fuzzy extension of f and {Ai : i ∈ I} be a family of multi-fuzzy subgroup of G1 . The multi-fuzzy set Ai is a multifuzzy subgroup of G1 if and only if f (Ai ) is a multi-fuzzy subgroup of G2 .

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Proof. Assume that Ai is a multi-fuzzy subgroup of G1 . Theorems 2.9 (c) and 3.4 together implies f (Ai ) = f (Ai ) is a multi-fuzzy subgroup of G2 . Conversely assume that f (Ai ) is a multi-fuzzy subgroup of G2 . Theorems 2.10, 2.9(h) and 3.5 together implies Ai = f −1 (f (Ai )) = f −1 (f (Ai )) is a multi-fuzzy subgroup of G1 . Theorem 3.7. Let G1 and G2 be groups, f be a surjective group homomor phism from G1 into G2 , a surjective order homomorphism h : M → Lj i −1 (that is, h(h (l)) = l, ∀l ∈ Lj ) be the bridge function for the multi-fuzzy extension of f and {Bj : j ∈ J} be a family of multi-fuzzy subgroup of G2 . The multi-fuzzy set Bj is a multi-fuzzy subgroup of G2 if and only if f −1 (Bj ) is a multi-fuzzy subgroup of G1 . Proof. Similar to theorem 3.6.

3.1

Normal multi-fuzzy subgroups

Definition 3.8. A multi-fuzzy subgroup A of a group G is called normal, if for each x ∈ G, A(x) ≤ A(gxg −1) g∈G

Theorem 3.9. Let A be a multi-fuzzy subgroup of a group G. Then the following conditions are equivalent for each x, g ∈ G : (i) A is a normal multi-fuzzy subgroup of G; (ii) A(x) ≤ A(gxg −1); (iii) A(x) = A(gxg −1 ); (iv) A(xg) = A(gx). Proof. Straight forward. Theorem 3.10. If A1 and A2 are two normal multi-fuzzy subgroups of a group G, then their intersection A1 A2 is a normal multi-fuzzy subgroup of G. Proof. By the theorem 3.2, A1 A2 is a multi-fuzzy subgroup of G. For every x, g ∈ G, (A1 A2 )(gxg −1) = A1 (gxg −1) ∧ A2 (gxg −1) = A1 (x) ∧ A2 (x) = (A1 A2 )(x). Hence A1 A2 is a normal multi-fuzzy subgroup of G. Theorem 3.11. A is a normal multi-fuzzy subgroupof a group G if and only if each level subgroup A[α] is normal in G, for α ∈ Mi .

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Proof. First we prove that, A is a multi-fuzzy subgroup of G if and only if each level subset A[α] is a subgroup of G. Assume that A is a multi-fuzzy subgroup of G. Let α ∈ Mi be arbitrary (fixed), for any x, y ∈ A[α] (that is, α ≤ A(x), α ≤ A(y)), we have α ≤ A(x) ∧ A(y) ≤ A(xy −1 ). Therefore xy −1 ∈ A[α] and implies A[α] is a subgroup of G. Conversely assume that A[α] is a subgroup of G, for each α ∈ Mi . Hence x, y ∈ A[α] implies xy −1 ∈ A[α] . That is α ≤ A(x) ≤ A(xy −1 ) and α ≤ A(y) ≤ A(xy −1 ). Therefore α ≤ A(x) ∧ A(y) ≤ A(xy −1 ) and hence A is a multi-fuzzy sub group of G. Finally, an arbitrary α ∈ Mi , A[α] is a normal subgroup of G if and −1 only if x ∈ A[α] implies gxg ∈ A[α] , ∀g ∈ G if and only if ∀x ∈ X, A(x) ≤ A(gxg −1) if and only if A is a normal multi-fuzzy subgroup of G. g∈G

Theorem 3.12. Let G1 and G2 be groups, f be a group homomorphism from G1 onto G2 and a meet preserving order homomorphism h : Mi → Lj be the bridge function for the multi-fuzzy extension of f . (i) If A is a normal multi-fuzzy subgroup of G1 , then f (A) is a normal multifuzzy subgroup. (ii) If B is a normal multi-fuzzy subgroup of G2 , then f −1 (B) is a normal multi-fuzzy subgroup. Proof. (i) For any x0 ∈ G1 , we have h(A(x0 )) ≤ h( A(gx0 g −1)) ≤

−1

h(A(gx0 g )), since A(x0 ) ≤

g∈G1

h(A(x)) ≤

x∈f −1 (y)

is f (y) ≤

g∈G1 −1

A(gx0 g ). Let y = f (x0 ), then

g∈G1

h(A(gxg −1)) ≤

x∈f −1 (y) g∈G1

h(A(gxg −1)). That

g∈G1 x∈f −1 (y)

f (A)(f (g)yf (g)−1). Hence f (A) is a normal multi-fuzzy sub-

g∈G1

group of G2 .

(ii) For any x ∈ G1 , f −1 (B)(x) = h−1 (B(f (x))) ≤ h−1 (

B(uf (x)u−1)).

u∈G2

Since f is onto, there G2 , such that u = f (g). Hence exists a g ∈ G1 for each u ∈ −1 −1 −1 f (B)(x) ≤ h ( B(f (g)f (x)(f (g)) )) ≤ h−1 (B(f (g)f (x)(f (g))−1)) = g∈G1

−1

g∈G1 −1

h (B(f (gxg ))) =

fuzzy subgroup of G1 .

g∈G1

g∈G1

f

−1

−1

(B)(gxg ). Hence f −1 (B) is a normal multi-

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4


Concluding remarks

In this paper we considered order homomorphism as bridge function for the multi-fuzzy extensions of functions. Order homomorphisms, lattice homomorphisms, arbitrary join preserving maps, complement preserving maps, etc. are useful bridge functions for multi-fuzzy extensions. ACKNOWLEDGEMENTS The authors are very grateful to Prof.T.Thrivikraman and referees for their very constructive comments. The first author is thankful to the University Grants Commission of India for awarding teacher fellowship under the Faculty Development Programme (No.F.FIP/XI Plan/KLKA001 TF 01).

References [1] P.S. Das, Fuzzy groups and level subgroups, Journal of Mathematical Analysis and Applications, 84 (1981), 264-269. [2] J.A. Goguen, L-fuzzy sets, Journal of Mathematical Analysis and Applications, 18 (1967), 145-174. [3] W.J. Liu, Fuzzy invariant subgroup and fuzzy ideal, Fuzzy Sets and Systems, 8 (1981), 133-139. [4] N.P. Mukharjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Information Sciences, 34 (1984), 225-239. [5] A. Rosenfeld, Fuzzy groups, Journal of Mathematical Analysis and Applications, 35 (1971) 512-517. [6] S. Sabu and T.V. Ramakrishnan, Multi-fuzzy sets, International Mathematical Forum, 50 (2010), 2471-2476. [7] S. Sabu and T.V. Ramakrishnan, Multi-fuzzy topology, International Journal of Applied Mathematics (accepted). [8] G.J. Wang, Order-homomorphism on fuzzes, Fuzzy Sets and Systems, 12 (1984), 281-288. [9] L. Ying-Ming, Structures of fuzzy order homomorphisms, Fuzzy Sets and Systems, 21 (1987), 43-51. [10] L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. Received: September, 2010